{"id":"7a15fab2-24da-4b85-a90c-6033e9ba3b67","arxiv_id":"1908.01670","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The two-loop QCD threshold corrections that mix strong and electroweak couplings in the MSSM Higgs-mass EFT calculation are computed for the first time, completing the two-loop QCD matching at the SUSY scale.","lead":"This paper computes the missing two-loop corrections that mix the strong and electroweak forces in the prediction for the Higgs boson mass in supersymmetric models with heavy superparticles. The new effects shift the predicted mass by about 100 MeV, small but comparable to how well the LHC can already measure the real Higgs mass.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the completion claim is internally consistent and the stated limitations are properly scoped.","rationale":"The reader's ACCEPT verdict is appropriate. The calculation is a technical two-loop EFT matching computation with strong internal consistency checks: the RGE check reproduces the known SM beta-function coefficient, the IR divergences cancel as required, and the WFR and DRED–DREG pieces are checked against independent literature. The explicit degenerate result Eq. (27) is sufficiently detailed to support the numerical claims, and a term-by-term summation of the displayed contributions is internally consistent. The reader's weakest assumptions (degenerate mass scale, heavy-gluino hierarchy, first-two-generation Yukawas) are genuine scope limitations but are explicitly flagged in the paper; they do not affect the algebraic completeness of the two-loop matching at a common SUSY scale. The main caveat is that the full generic-mass expressions are only available on request, which limits independent verification of the completeness claim beyond the degenerate limit; this justifies moderate rather than high confidence but does not warrant changing the verdict.","tokens_in":24008,"tokens_out":51528,"duration_ms":503707,"concrete_test":"Independently evaluate the degenerate limit by summing the displayed contributions Eqs. (12), (14), (16), (21), (23), (25) with m_Qi=m_Ui=m_Di=m_g=MS and F as in Eq. (24), and verify term-by-term agreement with Eq. (27). Additionally, request the electronic files from the authors and confirm that the generic-mass expressions reduce to Eq. (27) in this limit and reproduce the numerical shift in Fig. 2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The central claim that the mixed QCD–EW two-loop corrections complete the two-loop QCD matching of the quartic coupling is supported by the explicit degenerate-mass result Eq. (27), the RGE-consistency check Eq. (28), the cancellation of IR divergences between Eqs. (14)–(17), and the agreement of the WFR and DRED–DREG pieces with refs. [31–33]. I spot-checked the degenerate combination: with F(M_S,M_S)=7/12+(1/3)ln(M_S^2/Q^2), the sum of Eqs. (12), (14)–(17), (21), (23), (25) reproduces the pure-gauge and log structure of Eq. (27), including the absence of O(g^2 g'^2) terms. The numerical scenario assumes a common SUSY scale and neglects first-two-generation Yukawas; both are explicitly stated and do not affect the algebra of the correction, only the domain of the numerical application. The full generic-mass expressions are offered only on request, which limits immediate reproducibility but does not undermine the printed argument. No load-bearing technical objection found.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper computes the two-loop threshold corrections to the quartic Higgs coupling in the MSSM-to-SM EFT matching that involve both the strong and electroweak gauge couplings (the mixed QCD-EW corrections). Section 2 decomposes the correction into 1PI, wave-function-renormalization, and renormalization-scheme pieces, and gives a complete combined result in the degenerate superpartner-mass limit in Eq. (27). Section 3 compares two independent full two-loop computations of the SM relation between the pole Higgs mass and the MS-renormalized quartic coupling, refs. [85] and [86] (the code mr), finding agreement at the level of about 20 MeV near m_h = 125 GeV. Section 4 quantifies the numerical impact in a common-scale SUSY scenario: roughly -100 MeV on the predicted Higgs mass and about +100 GeV on the inferred stop mass scale. Section 5 discusses limitations, including the case where the gluino is much heavier than the squarks and the scenario would require a separate decoupling step.","tokens_in":24141,"tokens_out":6929,"duration_ms":78748,"significance":"If correct, the result closes the last missing two-loop strong-interaction contribution to the quartic-coupling matching in the single-scale MSSM EFT, thereby completing the two-loop QCD part of the threshold correction. The paper's main strengths are the clean decomposition of the calculation, the explicit degenerate-mass formula in Eq. (27), the nontrivial cross-checks against the independent self-energy results of refs. [31-33], the cancellation of IR divergences between Eqs. (14)-(17), the DRED-DREG shift, and the RGE-consistency check in Eq. (28). The reported numerical impact is modest and honestly assessed as subdominant to the gaugeless two-loop corrections, while being comparable to current LHC Higgs-mass precision. The stated limitations—the common-scale numerical scenario, the neglect of first-two-generation Yukawa couplings, and the gluino-mass-hierarchy caveat in Sec. 5—are explicit and do not undermine the algebra of the correction, though they bound the domain of the completeness claim.","major_comments":[],"minor_comments":[{"comment":"The full generic-mass results are described as available on request, but they are not included in the manuscript or as an ancillary file; I request that they be provided as supplementary material so that the completeness claim for arbitrary superpartner masses can be independently checked and reused.","section":"Sec. 2 and Sec. 5"},{"comment":"The phrase 'completes the calculation of the two-loop QCD corrections to the quartic coupling' should be explicitly qualified by the single-scale assumption under which the calculation is performed, given the paper's own discussion that a much heavier gluino would require an additional threshold-decoupling step.","section":"Abstract and Sec. 5"},{"comment":"The IR-divergent logarithm ln(-p^2/Q^2) should be accompanied by a standard +i0 prescription or an explicit statement of the convention used, since the expression as printed is ambiguous for p^2 < 0.","section":"Eq. (14)"}],"recommendation":"minor_revision","confidential_remarks":"This is a solid precision-calculation paper in the mainstream hep-ph EFT tradition. The scientific content appears correct and the cross-checks are convincing. The requested changes are presentational and reproducibility-related rather than scientific; I do not see a need for another full round of refereeing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this paper does what it says. It computes the mixed QCD–EW two-loop threshold corrections to the quartic Higgs coupling in the MSSM-to-SM EFT, and combined with the gaugeless-limit results of refs. [65,72] that completes the two-loop strong-interaction matching. The numerical effect is O(100) MeV on m_h and O(100) GeV on the inferred stop scale — modest but not negligible, and the authors say so plainly. The central derivation in Sec. 2 is cleanly organized into 1PI, WFR, and scheme-shift pieces, and it is cross-checked several independent ways: agreement with Martin’s self-energies in the right limits, cancellation of IR divergences in the matching, consistency with the SM beta-function via Eq. (28), and a sensible comparison of the two EW-scale computations in refs. [85] and [86]. I did not recompute the two-loop integrals, so my confidence is moderate rather than high, but I found no load-bearing flaw. The stress-test note is right: the degenerate-mass formula in Eq. (27) has the right structure, and the absence of O(g^2 g'^2) terms is expected from group theory. The soft spots are real but properly flagged. The full generic-mass results are only available on request, which limits immediate reproducibility — a fair criticism, not a fatal one. The numerical scenarios assume a common SUSY scale and neglect first-two-generation Yukawas; those are stated limitations that bound the domain of the numerical claims rather than the algebra of the correction. The gluino-heavier-than-squarks case is explicitly left to a separate decoupling step, which is honest. The citation pattern is the normal one for this field: heavy reliance on the authors’ own earlier work, but the new pieces are checked against independent published results, so the circularity burden is low. This is a specialist paper. It will be cited by people working on MSSM Higgs-mass predictions and EFT matching, and it deserves a serious referee. The main referee request should be: deposit the electronic results or make them available in a way that outlives the 'on request' mechanism.","headline":"Solid, complete-looking two-loop QCD threshold corrections that close a known gap in the MSSM EFT Higgs-mass program, with honest scoping of the numerical impact.","tokens_in":24838,"tokens_out":832,"would_cite":true,"duration_ms":10727,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper completes the two-loop QCD threshold corrections to the Higgs quartic coupling in the MSSM and finds the new terms shift the predicted Higgs mass by about 100 MeV.","keywords":["MSSM","Higgs mass","two-loop threshold corrections","quartic Higgs coupling","effective field theory","QCD corrections","strong-electroweak corrections","supersymmetry"],"falsifier":"Repeat the matching in a split scenario with, for example, $m_{\\tilde g}=10M_S$ and recompute the two-loop QCD-EW threshold correction including the gluino-mass-enhanced terms; if the predicted Higgs mass shifts by more than about 100 MeV relative to the degenerate-mass formula, or if the $\\beta$-function check in eq. (28) fails, then the completeness and numerical-impact claims are confined to the degenerate-mass scenario.","tokens_in":1806,"feed_emoji":"⚛️","tokens_out":3985,"duration_ms":133313,"temperature":0.7,"pith_summary":"The paper fills the last missing strong-interaction piece in the two-loop matching of the Minimal Supersymmetric Standard Model (MSSM) onto the Standard Model: the threshold corrections to the quartic Higgs coupling that involve both the strong gauge coupling and the electroweak gauge or Yukawa couplings. Together with earlier results obtained in the gaugeless limit, this completes all two-loop QCD corrections at the SUSY scale. The paper also compares two independent full two-loop computations of the relation between the quartic coupling and the pole Higgs mass at the electroweak scale, finding agreement to about 20 MeV near the measured Higgs mass. In a representative heavy-superpartner scenario, the new corrections lower the predicted Higgs mass by about 100 MeV and raise the common SUSY scale needed for a 125 GeV Higgs by about 100 GeV, an effect comparable to the LHC's measurement accuracy.","feed_headline":"New corrections complete two-loop QCD Higgs-mass matching","feed_subtitle":"Strong-electroweak threshold terms move the predicted mass by about 100 MeV, matching LHC precision.","key_machinery":"The central object is the two-loop mixed QCD-EW threshold correction $\\Delta\\lambda_{2\\ell,\\text{QCD-EW}}$ to the quartic Higgs coupling, which the paper splits into three pieces: two-loop one-particle-irreducible contributions from the effective potential, two-loop Higgs wave-function-renormalization contributions that require momentum-dependent self-energies, and a renormalization-scheme contribution that converts DR-renormalized MSSM couplings ($\\lambda$, $g$, $g'$, and the top Yukawa) to the MS-renormalized SM couplings. In the degenerate superparticle limit these combine into the explicit formula in eq. (27). The machinery carries the argument because the momentum-dependent self-energy and scheme-conversion terms are the parts that the earlier gaugeless calculations could not see, and they are also the parts that turn the completed SUSY-scale matching into a genuine next-to-next-to-leading-logarithmic resummation step.","core_discovery":"The central claim is that the class of two-loop threshold corrections of $O(g_{t,b}^2 g_s^2)$, $O(g^4 g_s^2)$ and $O(g'^4 g_s^2)$ to the quartic Higgs coupling has now been computed, so that, combined with the earlier $O(g_t^4 g_s^2)$ and $O(g_b^4 g_s^2)$ results, the two-loop QCD corrections at the SUSY scale are complete. The calculation goes beyond the effective-potential method used for the earlier gaugeless corrections: it requires the momentum-dependent two-loop self-energies of the Higgs and Z bosons, plus scheme-conversion terms between dimensional reduction and dimensional regularization and between MSSM and SM couplings. In the degenerate-mass limit the combined result is given by eq. (27), and it passes the consistency check of reproducing the SM $\\beta$-function term for the quartic coupling that is proportional to $\\lambda g_s^2(g_t^2+g_b^2)$. The paper further maintains that, in the EFT approach, the electroweak-scale determination of the pole Higgs mass should be performed at the same perturbative order, and shows that two existing full two-loop determinations agree to about 20 MeV near 125 GeV, within the quoted theoretical uncertainty of about 150 MeV.","pith_inferences":["The same computational pieces, momentum-dependent two-loop self-energies plus DR-MS scheme conversions, could be reused to compute the remaining electroweak-only two-loop matching corrections, which would complete the full two-loop matching rather than just its QCD part.","The numerical direction of the shift, toward a slightly heavier SUSY scale for the same Higgs mass, suggests that if the MSSM is realized with stops near the current LHC reach, the mixed QCD-EW corrections slightly tighten the region of parameter space compatible with 125 GeV.","A two-step EFT that decouples the gluino before the squarks would provide a direct test of the degenerate-scale assumption; in split-SUSY spectra the inferred value of $M_S$ could move by more than the quoted 100 GeV."],"forward_implications":["The heavy-MSSM prediction for $m_h$ no longer has any missing two-loop strong-interaction matching terms, so the remaining SUSY-scale uncertainty is set by uncomputed electroweak-only two-loop pieces and by higher orders.","In the degenerate scenario, the new corrections move the predicted Higgs mass down by about 100 MeV and move the common SUSY scale that yields $m_h = 125.09$ GeV up by about 100 GeV.","The electroweak-scale comparison shows that the two full two-loop calculations of the quartic-to-pole-mass relation agree to about 20 MeV near the observed mass, so that side of the EFT chain is not the dominant uncertainty.","This is a necessary step toward a full NNLL resummation of the logarithmically enhanced corrections to the MSSM Higgs mass.","If superpartners are discovered in the TeV range, the Higgs mass becomes a sharper precision constraint on stop masses and mixing, with theoretical and experimental uncertainties of comparable size."],"supporting_citations":[{"why":"Supplies the one-loop and gaugeless two-loop matching results and the conventions for the Higgs doublet rotation and squark-sector parameters that the new corrections complete.","marker":"[65]"},{"why":"Adds the remaining gaugeless two-loop Yukawa corrections and the DR-MS conversion for the bottom Yukawa coupling used in the numerical setup.","marker":"[72]"},{"why":"Provides one full two-loop computation of the relation between the quartic coupling and the pole Higgs mass at the electroweak scale, including the interpolating formula used in the comparison.","marker":"[85]"},{"why":"Provides the independent full two-loop electroweak-scale calculation whose public program is used to extract SM parameters and run them to the SUSY scale.","marker":"[86]"},{"why":"Supplies the matching-and-running engine that takes physical observables as input and returns MS-renormalized SM couplings at the SUSY scale.","marker":"[108]"}],"fun_headline_variants":["Two-loop QCD Higgs-mass corrections now complete","MSSM Higgs mass: two-loop QCD corrections finished","Threshold terms close two-loop QCD calculation","~100 MeV shift completes Higgs-mass matching","Full two-loop QCD corrections to heavy SUSY Higgs"],"cache_read_input_tokens":26880,"weakest_assumption_plain":"The quantitative results assume that all superparticles, including the gluino (the gluon's supersymmetric partner), and the heavy Higgs doublet share one common mass scale, with only the Standard Model below it; if the gluino is much heavier than the squarks, additional two-loop terms appear that the quoted 100 MeV numbers do not include.","fun_headline_variants_meta":{"raw":{"variants":["Two-loop QCD Higgs-mass corrections now complete","MSSM Higgs mass: two-loop QCD corrections finished","Threshold terms close two-loop QCD calculation","~100 MeV shift completes Higgs-mass matching","Full two-loop QCD corrections to heavy SUSY Higgs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000273,"raw_usage":{"total_tokens":1624,"prompt_tokens":925,"completion_tokens":699,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":541,"completion_tokens_details":{"reasoning_tokens":625}},"tokens_in":541,"tokens_out":699,"duration_ms":7307,"temperature":1.0,"reasoning_tokens":625,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:07:11.117417+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Repeat the matching in a split scenario with, for example, $m_{\\tilde g}=10M_S$ and recompute the two-loop QCD-EW threshold correction including the gluino-mass-enhanced terms; if the predicted Higgs mass shifts by more than about 100 MeV relative to the degenerate-mass formula, or if the $\\beta$-function check in eq. (28) fails, then the completeness and numerical-impact claims are confined to the degenerate-mass scenario.","supporting_citations":[],"review_version":1}